Question #40461

Let $f(z) = 12z^3 + k z^2 + c z$, where $k, c \\in \\mathbb{R}$. A particle moves in the complex plane along the curve $z(t) = e^{it}$ for $t \\in \\left[0, \\frac{\pi}{2}\\right]$. Let $u(t) = \\text{Re}(f(z(t)))$ and $v(t) = \\text{Im}(f(z(t)))$. It is given that $v(t)$ satisfies the conditions of Rolle's Theorem on the interval $\\left[0, \\frac{\pi}{2}\\right]$, and $u(t)$ has a local extremum at $t = \\arccos\\left(\\frac{3}{4}\\right)$. If $M$ and $m$ are the absolute maximum and absolute minimum values of $u(t)$ on $\\left[0, \\frac{\pi}{2}\\right]$, find the value of $8(M + m)$.